#### Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 3 Subquestion (iii).

Answer: $\left ( 2A^{T} \right )=2A^{T}$

Given: $A=\begin{bmatrix} 1 & -1 &0 \\ 2&1 & 3\\ 1 &2 & 1 \end{bmatrix}, B=\begin{bmatrix} 1 & 2 &3 \\ 2 & 1&3 \\ 0 & 1 & 1 \end{bmatrix}$

To prove: $\left ( 2A^{T} \right )=2A^{T}$

Hint: The  $A^{T}$ of matrix  $A$ can be obtained by reflecting the elements along it’s main diagonal.

Solution:

$A^{T}=\begin{bmatrix} 1 & 2 &1 \\ -1 & 1& 2\\ 0 &3 & 1 \end{bmatrix}$

$\left ( 2A^{T} \right )=2A^{T}$

$\left ( 2\begin{bmatrix} 1 & -1 & 0\\ 2& 1 &3 \\ 1 & 2 & 1 \end{bmatrix} \right )^{T}=2\begin{bmatrix} 1 &2 &-1 \\ -1 &1 & 2\\ 0 &3 & 1 \end{bmatrix}$

$\begin{bmatrix} 2 & -2 & 0\\ 4& 2 &6 \\ 2 & 4 & 2 \end{bmatrix}^{T}=\begin{bmatrix} 2 &4 &2 \\ -2 &2 & 4\\ 0 &6 & 2 \end{bmatrix}$

$\begin{bmatrix} 2 &4 &2 \\ -2 &2 & 4\\ 0 &6 & 2 \end{bmatrix}=\begin{bmatrix} 2 &4 &2 \\ -2 &2 & 4\\ 0 &6 & 2 \end{bmatrix}$

∴LHS=RHS

$\left ( 2A^{T} \right )=2A^{T}$